On the \(D_\alpha\) spectral radius of non-transmission regular graphs

Let \(G\) be a connected graph with order \(n\) and size \(m\). Let \(D(G)\) and \(Tr(G)\) be the distance matrix and diagonal matrix with vertex transmissions of \(G\), respectively. For any real \(\alpha\in[0,1]\), the generalized distance matrix \(D_\alpha(G)\) of \(G\) is defined as $$D_\alpha(G)=\alpha Tr(G)+(1-\alpha)D(G).$$ The \(D_\alpha\) spectral radius of \(G\) is the spectral radius of \(D_\alpha(G)\), denoted by \(\mu_{\alpha}(G)\). In this paper, we establish a lower bound on the difference between the maximum vertex transmission and the \(D_\alpha\) spectral radius of non-transmission regular graphs, and we also characterize the extremal graphs attaining the bound.